Sorry for multiple postings but I'm still having this issue. Another thing I tried was turning my wireless router off for 30 seconds and then turning it back on. Still no results.
I also noticed that when I tried playing things in the iTunes store on another computer, signed into another account, I still have this problem.
This is telling me that it may be a problem with my network, or a problem with the latest iTunes update, 10.6.3. I've got Verizon FiOS, and I don't have any new devices that could be interfering with the network. I think my next step would be to install iTunes 10.6.1, and see if that helps. I'll report back when I'm done.